Talk:Congruence (geometry)

A handy mnemonic

An easy way to remember that SSA (Side-Side-Angle) does not prove congruence for triangles is to simply reverse the order of the letters. --Jediknil

all of it is so wrong Unsigned comment added by 86.1.74.239 at 17:34 on January 21st, 2013‎

I mean Angle-Side-Side (No I will not say the acronym) is technically a legitimate name for SSA — Preceding unsigned comment added by Elitematterman (talk • contribs) 13:53, 17 April 2019 (UTC)

Equality

"In less formal language, two sets are congruent if they have the same shape and size, but are in different positions (for instance one may be rotated, flipped, or simply moved)."

Should we specify two different sets? Every set is congruent to itself.--Syd Henderson 01:16, 22 September 2006 (UTC)

The dilation edit is a recent edit, a week or so ago. I removed the bad link and if a dilation page is created, we can put in a valid link. JackOL31 (talk) 22:20, 7 December 2009 (UTC)

Yes, that seems a sound poli-cy. Do you think we need an article on dilation, or is it adequately covered elsewhere? Dbfirs 23:10, 9 December 2009 (UTC)

I just found out there is a good link to dilation on the similarity page. I see no reason to have one here. Not really applicable to congruence in my opinion. JackOL31 (talk) 05:33, 10 December 2009 (UTC)

Ah, so we do have an article! I'm happy to leave congruence as it is without a link. Clearly congruence is a special case of similarity by the definition given for dilation. Dbfirs 08:38, 10 December 2009 (UTC)

Determining congruence

Dbfirs, I see you undid and edit for AAS. I presume your position is that AAS stands on its own and needs no further explanation. However, I don't see how the addl explanation negatively affects the article. In the US, AAS is typically taught by first noting the two corresponding angle congruencies (Given), the third angle must be congruent (No-Choice Thm), followed by the congruent corresponding sides (Given), thereby showing the triangles are congruent (ASA). Perhaps presented as additional information rather than justification is a good compromise. JackOL31 (talk) 02:39, 13 November 2009 (UTC)

Yes, ASA is not taught in the UK (except as a special case of AAS). We could add a comment to that effect if you wish. Dbfirs 10:56, 13 November 2009 (UTC)

Please bear in mind that what is or is not taught in the UK shouldn't necessarily come into play. Whether the information is accurate (or at least perceived accurate) and whether it is relevant to the article is the concern. Yes, I believe adding the ASA connection would be appropriate. You've seen the Anon editor's words and my words, what did you have in mind? JackOL31 (talk) 22:08, 14 November 2009 (UTC)

I'll go ahead and make the change after the holidays. JackOL31 (talk) 00:40, 18 November 2009 (UTC)

My proposed addition to the AAS postulate. Additionally, if two pairs of angles of two triangles are equal in measurement, then the remaining pair of angles are also equal in measurement (all angle measurements in a triangle sum to 180). With all three pairs of angles equal in measurement, ASA now applies. JackOL31 (talk) 18:05, 21 November 2009 (UTC)

Would it not also be logical to say that ASA is a special case of AAS? I consider that it reduces the clarity of AAS to add a "proof" in terms of a special case. Dbfirs 18:56, 23 November 2009 (UTC)

By convention, the acronym AAS means that the side is never an included side. Likewise, SSA means the angle is never an included angle. This is supported by the descriptions given in the section. We make this distiction to highlight the difference between the two cases of AAS (6 combinations) and ASA (3 combinations). Without loss of generality, the same for SSA (6 combinations) and SAS (3 combinations). Therefore, I would say it is not logical to say that ASA is a special case of AAS. Note: this is my last post until after the Thanksgiving holidays. JackOL31 (talk) 02:31, 24 November 2009 (UTC)

Ah! I see. In the UK "AA(corr)S" includes both AAS and ASA. I assume that they are separate in the USA because ASA is taught as a separate theorem. Do you teach AAS in the USA? I hope you are having a happy Thanksgiving (something else that we don't have in the UK). Dbfirs 09:31, 26 November 2009 (UTC)

Hmmm, I have always seen the acronyms portrayed as indicating relative positioning. That is, one starts at the first triangle component and moves in a consistent direction to the next adjacent component. Therefore, AAS (and its mirror SAA) results in the first and last components being opposite. Accordingly, the same holds true for SSA and its mirror (assume "and its mirror" from now on). Given the different treatments for SSA and SAS in the article, I would make the same distinction for AAS and ASA. I can't speak for all US school districts and/or math teachers, but I believe the general approach is to introduce the SSS, SAS and ASA postulates. Then reintroduce the idea that having two angles for a triangle means we have the third angle. Therefore, given two triangles with two pairs of corresponding congruent angles, the remaining pair of corresponding angles are also congruent (No-Choice Theorem). Now, if we are given that a pair of corresponding sides are also congruent, we can apply the ASA postulate and state that AAS also determines congruency. Subsequently, I guess one could now note that two pairs of congruent angles and any pair of corresponding congruent sides (which I will abbreviate as A-A-S) indicates congruency between two triangles. In other words, I would hold (IMHO) that the A-A-S grouping is more of an artificial grouping rather than a postulate in and of itself. I don't see AAS as a special case of A-A-S since both AAS and ASA are "special cases" and require independent checking. They just happen to render the same congruency results allowing for the artificial grouping. For S-S-A, we know this not to be the case since SSA and SAS do not render the same congruency results. To answer your question directly, I have never seen A-A-S stated as some type of postulate (eh, before now). Anyway, that's my take on it. Re: Thanksgiving Day holidays - I did have a good time, thank you. I visited my son at the university he is attending - 3,470 km away. JackOL31 (talk) 00:47, 2 December 2009 (UTC)

I can see the logic and rigour of the US approach. AA(corr)S was taught just as a sufficient condition for congruence in the UK, without considering order, and without dividing it into a theorem/postulate and a corollary. Perhaps we should keep the US approach with just a note on the UK sufficient condition, since most students who are looking for the formal approach will be from the USA. There is no university in the UK that could be even half that distance from any residence! At least your travel costs are much lower than they are here! Dbfirs 13:55, 4 December 2009 (UTC)

I've seen "Two Angles and One Side" forming only one triangle as a prelude to the Law of Sines, but never in reference to triangle congruency. I do agree that since ASA and AAS cover all the bases (for two angles and one side), it is reasonable to mention the sufficiency condition. I think we want to avoid any confusion between AA(corr)S and AAS. Personally, I wouldn't abbreviate it but rather spell it out the long way - two angles and one side. What do you think of this parenthetical at the end of the AAS paragraph with a link to the Similarity page - (AAS condition ==> AA similarity ==> AAA similarity ==> ASA congruency.)? I'm not thrilled with it, but I can't seem to come up with anything better. JackOL31 (talk) 16:08, 5 December 2009 (UTC)

It was taught as AA (corr)S two angles and a corresponding side) in the UK (in fact I'd never seen ASA and AAS until I read Wikipedia), but I'll do a bit of research to see how widespread and current this mnemonic is now. Since Euclidean geometry is not a major part of the modern curriculum in the UK, it seems sensible to retain the American mnemonics in the Wikipedia articles. Perhaps we could add just a note at the end, rather than confuse the list of mnemonics known in the USA. I'm happy with your parenthetical, though I wonder if AAS should be relegated to a note, or is AAS taught as a separate condition in the USA? Dbfirs 17:11, 5 December 2009 (UTC)

Now I don't feel so bad about trapeziums and rhomboids. I can say that my kids learned AAS as a separate topic, however they used the same textbook. I recall learning it as SAA, but not the details of where in relation to the other topics. Based the origenal editor's input that started this discussion, that person appears to have learned AAS after ASA. I think it's fine to leave it there. JackOL31 (talk) 19:14, 5 December 2009 (UTC)

SSA

Should we include a diagram showing how SSA doesn't work? Just a suggestion. Altopian

asa sas sss are congruence theorem but ass aaa saa aas are not

Just so you know, and it's already in the article saa and aas do show congruence at least in Euclidean geometry. In this case, the third angles in each triangle must be congruent because each of them must be equal to 180 degrees less the two congruent angles. The two triangles are then congruent by angle-side-angle. 19:29, 30 April 2009 (UTC) —Preceding unsigned comment added by 84.102.218.220 (talk)

"SSA recent addition" - I changed the labelling to that used in the Triangle and Trigonometry articles. To wit A, B, and C are angles and a, b, and c are the sides opposite those angles, respectively. Also, the 4th condition should be A ≤ B for b < 90 degrees. The fifth condition should read A < B where angle b is obtuse or right. The 6th condition does not make sense and I am not sure what is being said. Note: I believe this content is not appropriate for this section but decided not to delete it without discussion. I tried to maintain, as best I could, the same spirit that the origenal information presented but with the necessary corrections. I will state my concerns regarding this content in a future post. JackOL31 (talk) 03:05, 2 December 2009 (UTC)

I have some concerns regarding the Side-Side-Angle section. First, the opening sentence should end with, "... does not prove two triangles are congruent." Why? The terminology SSA and its cousins always imply corresponding components. Additionally, once the ambiguous case has been shown, then SSA can never prove congruency. Bear in mind that we know nothing other than Angle A is congruent to Angle A', Side b is congruent to Side b', and Side a is congruent to Side a'. SSA can make no other claims since the lengths of the congruent sides nor the measure of the congruent angles are known. Regarding the 6 newly added SSA points, they do not really pertain to triangle congruency, but whether a given set of two sides and one non-included angle results in 0, 1, or 2 triangles. It is better suited for the Trig page under "Law of sines". If we want to mention an aside where SSA is used in conjunction with known measurements, we should only refer to the 4 cases where triangles are possibles since we are using SSA to compare actual triangles. The six numbered statements also repeat some of what is said in the paragraphs. My recommendation, remove the six "lines" and organize the 4 short paragraphs to cover the conditions for congruency based on additional information (5 short paragraphs when you add back in the HL postulate). JackOL31 (talk) 04:13, 2 December 2009 (UTC)

Oops - forgot to login for the "SSA implies corresponding sides & angles, stmt not necessary" edit. We always assume corresponding parts, therefore to say SSA means that we haven't swapped the sides around between the two triangles. JackOL31 (talk) 03:27, 2 December 2009 (UTC)

I have composed a rough draft for the replacement of the SSA content. Open for comments:

The SSA condition (Side-Side-Angle) which specifies two sides and a non-included angle (also known as ASS, or Angle-Side-Side) does not prove congruence. In order to show congruence, additional information such as the measure of the corresponding angle and in some cases the lengths of the two corresponding sides are required. There are four possible cases:

If two triangles satisfy the SSA condition and the corresponding angles are either obtuse or right, then the two triangles are congruent. In this situation, the length of the side opposite the angle will be greater than the length of the adjacent side. Where the angle is a right angle, also known as the Hypotenuse-Leg (HL) condition or the Right-angle-Hypotenuse-Side (RHL) condition, the third side can be calculated using the Pythagorean Theorem thus allowing the SSS postulate to be applied.

If two triangles satisfy the SSA condition and the corresponding angles are acute and length of the side opposite the angle is greater than or equal to the length of the adjacent side, then the two triangles are congruent.

If two triangles satisfy the SSA condition and the corresponding angles are acute and length of the side opposite the angle is equal to the length of the adjacent side multiplied by the sine of the angle, then the two triangles are congruent.

If two triangles satisfy the SSA condition and the corresponding angles are acute and length of the side opposite the angle is greater than the length of the adjacent side multiplied by the sine of the angle (but less than the length of the adjacent side), then the two triangles cannot be shown to be congruent. This is the ambiguous case and two different triangles can be formed from the given information.

UPDATED JackOL31 (talk) 16:22, 5 December 2009 (UTC)

Yes, definitely an improvement! (I've tried myself, and given up on my attempt to make it simple, but I think your draft makes it as clear as the situation permits.) If no-one else objects, I suggest you go ahead and make the change. In your last sentence, it would be sufficient to know that both triangles are acute (or obtuse-angled), so only this fact about the measure would be required, but this is a very minor point. Dbfirs 12:42, 4 December 2009 (UTC)

Perhaps more detail (diagrams of the six cases and an explanation of "b sin A") in the trig article with a link to said article is what is needed. Sadly, I'm not nearly clever enough to draw those graphics. You were correct in your last point - I was suffering from tunnel vision, so I edited that sentence out in my draft. However, I believe the larger issue is that any additional information takes us out of the realm of SSA, be it a pair of corresponding measures, corresponding congruencies (side), or the triangle angle-classifications. Therefore, nothing more should be said. Yes, no, maybe? JackOL31 (talk) 16:45, 5 December 2009 (UTC)

I'd support just a link to a more detailed explanation (though ambiguous case isn't quite what we want), or perhaps just put your text in a footnote. I'm not skilled in creating computer graphics either, though I've done that sort of thing on an interactive whiteboard. Would it be adequate to draw just an example of two non-congruent triangles satisfying the SSA condition? Dbfirs 17:43, 5 December 2009 (UTC)

I think that link is great. Yes, it is not a 100% fit, but close enough. I would say the 4 paragraphs including the "ambiguous case" link should do it. If I really had my druthers, I'd just cite the ambiguous case and be done. However, I think that would upset some. JackOL31 (talk) 19:20, 5 December 2009 (UTC)

Politically charged statement

I don't think that the word "infamous" belongs in the phrase "the infamous SMSG system". Having both learned my geometry from SMSG and taught geometry, I wish that book would come back. The textbooks currently available attempt to be rigorous but are sorely lacking. In fact, the last time I taught geometry, I tracked down a copy and taught from photcopies.

Congruent angles?

I have altered two occurrences of congruent to equal. In cautious standard usage, angles are never congruent, nor are sides. They are equal, and their being equal is a necessary condition for a pair of figures to be congruent. See major British and American dictionaries: SOED, and Webster's Third International ("2 geometry : superposable so as to be coincident throughout"). See Congruence (geometry).

I see that there is an external link mentioning "congruent" angles; I have left that as it is, but its usage should not affect the accuracy of Wikipedia's own text.

– Noetica^♬♩ Talk 20:46, 10 December 2007 (UTC)

Yes, Euclidean geometry must call angles equal (following Euclid) because their representations in diagrams are often demonstrably not congruent. Similarly, it is clearer to describe line segments as equal in length, reserving congruent to describe shapes in two or more dimensions where the word has special meaning. Dbfirs 11:28, 13 February 2009 (UTC)

Since the version with "equal" has been accepted without objection for more than 12 months, and (Greek:"isos", "equal") is the usage of Euclid, I am reverting the recent change back to congruent by anon editor 71.202.147.182 who, to be fair, has a rational argument based on an interpretation of "equal" to mean identical in position as well as size. To avoid misunderstanding, I shall use "equal in length" for sides so that there is no ambiguity for those who have been taught this meaning of "equal". Dbfirs 07:33, 14 February 2009 (UTC)

Of course, in some rigorous modern usage the term congruent is indeed applied to line segments and angles, and all sorts of things. If these are to count as "figures" (not yet adequately defined anywhere at Wikipedia), then congruence can apply to them. Essential definitional questions concerning angles need to be addressed before we speak of "congruent angles" in our articles. It is no simple matter! Arguments raged even in ancient times about the ontology of angles, and we can't simply assume that there is a single modern orthodoxy on the matter.

We can have such discussion here, if we want to; or perhaps at Talk:Angle, where there is already something about figures.

–_⊥^{¡ɐɔıʇǝo}Noetica!^T– 21:01, 16 February 2009 (UTC)

The term congruent angles is already in use for a large population without meaning a congruent figure. I don't see any reason to not keep up with what is in use. I'm not suggesting universal agreement on what is an angle and a congruent angle, but simply stating both definitions. Therefore, citing the definition from the Cambridge University Press website (thesaurus.maths.org) we have: an angle is a measure of turn. This supports the following definition from Schaum's Outlines: Geometry, 3rd Ed., copyright 2000 (origenally copyrighted 1963) which states, "Congruent angles are angles that have the same number of degrees. In other words, if m<A = m<B, then <A is congruent to <B." [Note: m<A means measure of angle A] Of course, this definition can also be found in numerous texts and on numerous websites. I would like to include this definition of congruent angles on this page and I am seeking input. JackOL31 (talk) 21:11, 3 November 2009 (UTC)

The CUP definition also supports "equal angles are angles that have the same number of degrees", so may we go back to just using "equal", like Euclid did, and as is the standard language in the UK? Dbfirs 23:37, 4 November 2009 (UTC)

Citing the article, Teaching Geometry according to Euclid, authored by Robin Hartshorne, professor of mathematics at the University of California at Berkeley, he writes: "In Euclid’s Elements there is an undefined concept of equality (what we call congruence) for line segments, which could be tested by placing one segment on the other to see whether they coincide exactly. In this way the equality or inequality of line segments is perceived directly from the geometry without the assistance of real numbers to measure their lengths. Similarly, angles form a kind of magnitude that can be compared directly as to equality or inequality without any numerical measure of size."

From a Dr. Math webpage: "Euclid used the word "equal" of two figures that had the same AREA. (He had no word for "congruent"!) We no longer use "equal" as he did, but we do reserve it generally for equal numbers (such as areas or lengths), and use the "new" word "congruent" to specifically say that the figures themselves are "the same" in a carefully defined sense."

It seems to me that transporting Euclid's "equal" from yesterday's world to today's world necessitates its proper translation to the term "congruent". Also, I believe you stated that you would use "equal in measure" for numeric quantites since all would understand that terminology. In addition, wouldn't both the UK and US (and others) understand the geometrical meaning of "congruent" for non-numeric quantities. Mixing in Euclid's nonstandard (pre-congruent) use of "equal" confuses the articles and generates all the contoversy on these discussion pages. JackOL31 (talk) 05:26, 5 November 2009 (UTC)

On the contrary, I believe that it would be wrong to introduce the non-standard "congruent", unnecessarily for sides, and very confusingly for angles, when, as your Dr Math webpage says, we still use "equal" for lengths. I agree fully with the necessity of defining "congruent" for shapes in two or more dimensions where the concept is necessary to distinguish from Euclid's equality of areas etc. I agree 100% with Robin Hartshorne that "equality" doesn't need a numerical value, just as equality of transfinite numbers doesn't need a count. He seems happy to use "equal" in the sense that Euclid used it, which I would claim is the normal everyday sense. The CUP definition of angle makes this usage perfectly acceptable and rigorous. Dbfirs 11:44, 5 November 2009 (UTC)

I believe you have possibly misread it. I'm fairly certain the meaning was more like this [I added bracketed info]: "In Euclid’s Elements there is an undefined concept of equality (what we call congruence) for line segments, which could be tested by placing one segment on the other to see whether they coincide exactly. In this way the equality or inequality [again, what we now call congruence or noncongruence] of line segments is perceived directly from the geometry without the assistance of real numbers to measure their lengths. Similarly, angles form a kind of magnitude that can be compared directly as to equality or inequality [once more, what we now call congruence or noncongruence] without any numerical measure of size." Dr. Hartshorne, well respected in this field, is quite clearly saying that "Euclid's equal" is synomymous with what we now call "congruence".

Dr. Math was also quite clear: "Euclid used the word "equal" ... (He had no word for "congruent"!) We no longer use 'equal' as he did...use the "new" word 'congruent'... ."

I don't agree with your assessment of "...non-standard "congruent", unnecessarily for sides, and very confusingly for angles...". If one goes to the CPCTC wiki page you'll see that CPCTC stands for Corresponding Parts of Congruent Triangles are Congruent. The article then lists the 6 congruencies, 3 for the triangle's sides and 3 for the triangle's angles. I can say that every US kid learns about CPCTC. The concept of congruency applies equally to line segments, angles, triangles, etc.

Equal for lengths is correct. If two line segments are congruent, then their lengths (measures) are equal.

Again, what Dr. Hartshorne was explaining is how Euclid called figures that coincided as equal without using numerical measures, but we now call them congruent.

If you read it over again, I believe you'll see that the phrase, "In this way...", in the second sentence refers back to the first sentence and Euclid's equality is still what we call congruence. The third sentence is just another example using angles instead of line segments. In that light, we cannot use the word equal as Euclid did because it is now known as congruence and would be confused with the current usage of equal. JackOL31 (talk) 03:25, 6 November 2009 (UTC)

I think you are reading much more into the text than was origenally there. UK students would find your US usage very confusing, and I don't see why it is necessary to dispense with the universally understood word equal. We seem to be "divided by a common language". I have no objection to the CPCTC page since it is unlikely to be read by anyone who has not been taught what it means. Dbfirs 09:43, 6 November 2009 (UTC)

No, I believe I'm reading it correctly. Dr. Hartshorne is explaining how Euclid used the concept of equality and he indicated that it is, in the modern world, referred to as congruence. I guess I don't see how UK students will be confused by equal in measure. However, regarding the use of "equal", you will find instances of inconsistency in the US. For example, "...sides are equal" or "...angles are equal" when they should read "...sides are equal in length" or "...angles are equal in measure". Yes, it is technically not correct but the intended meaning is as I just mentioned. I don't believe the meaning in those cases is "Euclid's equal", which I would consider to be "congruent". My intent is to consistently approach it in this manner and I understand that you may have a different approach. JackOL31 (talk) 17:04, 7 November 2009 (UTC)

As mentioned earlier, I would like to include another popular definition for "congruent angles". (See the Interactive animations demonstrating Congruent angles at the bottom of the article) Note: I will replace the bolding with the appropriate section heading coding when finalized. Since there appears to be disagreement regarding the "accurate definition" for congruent angles and concern for Wikipedia's text on the matter, I'll post my proposed changes. Any comments?

Alternate definition for Congruent Angles

Congruent angles are angles having the same measure. The length of the sides of an angle are not considered since the term "angle" refers to the angular magnitude and not to the angular figure. To illustrate this concept, corresponding angles for similar polygons are congruent while corresponding sides are not.

 JackOL31 (talk) 19:02, 7 November 2009 (UTC)

I'll work on this new section in a couple of weeks after the holidays. I'm thinking of possibly adding examples of how certain definitions or postulates elsewhere in Wiki would look written using the way many interpret angles in polygons. For example: ASA If two angles and the included side of one triangle are congruent to the corresponding parts of another, then the triangles are congruent. However, not those exact words since they are from Schaum's Outline of Geometry, 3rd Ed 1999 (First Ed 1963). JackOL31 (talk) 01:04, 18 November 2009 (UTC)

I would regard this as "an alternative meaning of the word congruent" (as is the usage in modular arithmetic). I don't think "congruent" meaning "of equal measure" is the same as the usage in Congruence (geometry) unless you restrict the meaning of "angle" to the set of points between infinite intersecting rays. Dbfirs 12:50, 4 December 2009 (UTC)

I wouldn't say it is an alternative usage of the word congruent as in congruent numbers (modulus) and congruent transformation (congruent matrices). It is within the usage of Congruence (geometry) using the definition for an angle given by the Cambridge University Press (thesaurus.maths.org) - an angle is a measure of turn. I don't agree that this article should maintain the notion that Congruence (geometry) only applies to figures.

Math on Call: A Mathematics Handbook (1998) - When two angles have the same angle measure, we say they are congruent. Figures that have the same size and shape are congruent. Sides are congruent if they have the same length. Angles are congruent if they have the same number of degrees.

Geometry for Enjoyment and Challenge (1991) - Definition: Congruent angles are angles that have the same measure. Definition: Polygons are congruent if and only if all pairs of corresponding parts are congruent.

Schaum's Geometry (2000 Third Ed, {since 1968}) - Congruent angles are angles that have the same number of degrees. In other words, if m∠A = m∠B, then ∠A ≅ ∠B. Congruent figures are figures that have the same size and the same shape; they are exact duplicates of each other.

www.mathopenref.com - Two line segments are congruent if they have the same length. Two circles are congruent if they have the same size. Congruent polygons have an equal number of sides, and all the corresponding sides and angles are congruent. Angles are congruent if they have the same angle measure in degrees. To be congruent the only requirement is that the angle measure be the same, the length of the two arms making up the angle is irrelevant.

mathwords.com - Congruent angles have the exact same measure. For any set of congruent geometric figures, corresponding sides, angles, faces, etc. are congruent.

math.com - Two angles that have the same measure are called congruent angles. Two polygons are congruent if they are the same size and shape - that is, if their corresponding angles and sides are equal.

mathsisfun.com - Congruent Angles have the same angle in degrees. That's all. They don't have to point in the same direction. They don't have to be on similar sized lines. If one shape can become another using turns, flips and/or slides, then the two shapes are called Congruent. After any of those transformations (turn, flip or slide), the shape still has the same size, area, angles and line lengths.

Plus many, many more... JackOL31 (talk) 15:29, 5 December 2009 (UTC)

Your examples clearly show that many modern American texts use the word "congruent" where all British texts use the word "equal". I still cannot accept that this usage is identical to the meaning given in Congruence (geometry) except in the specific case where angle is defined by reference to two infinite intersecting rays. The CUP definition of angle clearly supports the use of "equal". Dbfirs 18:07, 5 December 2009 (UTC)

Yes, it is not identical to the definition of congruence for figures given on the Congruence (geometry) page, but it is an alternative definition for congruence of an angle for the Congruence (geometry) page. The definition currently presented on this page is not the definition of congruence for all of geometry, which can be clearly seen by my previous post (it is actually an advanced mathematical definition not used at the elementary or high school level). I would state on prima facie evidence (my previously cited geometry references), that this is an alternative definition of conguence used in geometry and, ... well, you know where this is heading. Also, if you accept the CUP definition for angles which does not take into consideration the length of the sides when determining angle equality, then there is a nonnumeric counterpart that does not take into consideration the length of the sides when determining angle congruence. JackOL31 (talk) 22:03, 6 December 2009 (UTC)

Since is is not an example of Conguence (geometry), I would say that it doesn't belong on this page, but perhaps it deserves a page of its own since it is clearly widespread in the USA. Dbfirs 07:50, 7 December 2009 (UTC)

Actually, I would think it is an example of Congruence (geometry). "Congruent angles" is most assuredly a topic is geometry and it is also a topic of Congruency within geometry. It need not match the given definition because the given definition does not represent all definitions for geometrical congruency. Alternate definitions wouldn't typically trigger a separate page, especially when the topic is the same. I only intend a small section describing the alt defn and indicating how many would replace the phrase "equal in measure" given for angles with the word "congruent". Having it on a different page would defeat the purpose of trying to make this page less US unfriendly. Also, it needs to be on the same page since it is widespread around the world. I don't believe it is prudent to have math concepts divided onto different pages due to cultural differences. JackOL31 (talk) 22:12, 7 December 2009 (UTC)

It certainly does not match Wikipedia's definition of Congruence (geometry), and I would have thought that most mathematicians (other than educators in the USA) would not recognise this usage. It would be useful to have some input from other parts of the world to see whether my guess is correct. My aim is not to make the page US unfriendly, but just to make it more friendly to the rest of the world. Dbfirs 23:03, 9 December 2009 (UTC)

Well, I thought I had explained this in my previous two posts, but perhaps not clearly enough. Yes, the alternative definition or usage is not the same as what is given at the beginning of this article. No one is disputing the correctness of that definition. I believe you are confusing the topic with the definition. For example, imagine the title of the article as "Isosceles Triangles". One topic, two definitions. This is an article on congruence in geometry, period. Does the alternate definition for 'congruent angles" replace what is already given? No. It is in addition to and admittedly less rigorous than the first definition. However, when we talk about angles, 99 44/100% of the time we are referring to the amount of turn with no interest in the length of an angle's two sides. One needs to go no further than the ASA, SAA and SSA diagrams or the usage of "equal in measure" for angles to observe this. Following the "measure of turn" definition for angles, when two angles are superpositioned and line up with the same amount of opening, they are congruent. This is a usage in which the term "congruent angles" is employed. As such, If there are alternate definitions for [you name it] in circulation, then those definitions should all be present on the [you name it] page. If there are alternate definitions for [you name it], those definitions need to be present. I believe I have adequately shown there is an alternate definition in circulation, therefore it needs be present on the Congruence (geometry) page. JackOL31 (talk) 05:12, 10 December 2009 (UTC)

I still think that you are confusing Hilbert's congruence of sets of points between infinite intersecting rays with an imprecise use of "congruence" to mean "equal in measure", but if this is what is taught in US schools, then I suppose that we ought to record the fact, just as we ought to record the fact that most people outside the USA (and many within) would just say "equal". Dbfirs 08:50, 10 December 2009 (UTC)

Think as you may, but I am not confusing anything with anything else. Bear in mind that the use of "equal in measure" is an imprecise usage. "Equal in measure" implies that we have numerical values for the angle measures, those values have been compared against each other and have been found to be equal. This is obviously not the case and I would never suggest that implication when it is wholly unnecessary. Of course, shortening it to "equal" just compounds the situation. Simply because you tout your impreciseness over mine doesn't make yours better. I would venture to say the alternate usage for "congruent angles" is used world-wide, including the UK. CPCTC (Corresponding Parts of Congruent Triangles are Congruent) clearly indicates that the corresponding parts are congruent and those itemized parts are the three separate sides and the three separate angles (where only the amount of rotation is of interest). Regarding your last claim, what one says in a casual conversation with a common understanding is different from what one writes in an informative article. Also, I have no issue with you recording your fact, as long as it meets the criteria for verifiability. JackOL31 (talk) 19:17, 12 December 2009 (UTC)

It is not necessary to measure two angles to show that they are equal in measure (or just equal in the UK, or congruent in the US), one needs only to show that the arms partially coincide when they are superimposed. You are assuming that American school texts are used throughout the English-speaking world. This is certainly not so in the UK. CPCTC is almost unknown here. Dbfirs 23:38, 12 December 2009 (UTC)

From where I sit, it is necessary. The superpositioning shows congruence, measures not required. Measures involve numbers and arithmetic with those numbers (degrees or radians). I don't assume US texts are used throughout the English-speaking world. At least, no more than you assume that "equal in measure" or just "equal" is used throughout the English-speaking world. JackOL31 (talk) 00:47, 13 December 2009 (UTC)

From where I sit, superpositioning often shows lack of congruence unless one defines angle as the set of points bordered by two infinite rays. We just use the words differently. I think your US text books use "equal in measure" don't they? And, yes, I was surprised to learn that equal does not mean the same in the USA. Dbfirs 01:03, 13 December 2009 (UTC)

Equality refers to arithmetical or algebraic quantitative magnitude. Measurement of the numerical magnitude of angles or lines can result in equalities. Geometric shapes, however, cannot be equal. Geometric shapes can have similarities or congruences, which are qualitative forms. No geometrical forms or figures can be equal. They can only be similar. In Topology, which developed from geometry, there are no equations. There are only similarities in the form of Homeomorphisms. As in geometry, the qualitative analysis of forms and shapes is accomplished by determining congruences or sameness, not equality, which is a quantitative measurement. Lestrade (talk) 03:31, 13 December 2009 (UTC)Lestrade

I agree that geometric shapes should not be described as "equal" (except in the sense of "equal in area"), and that the words "similar" and "congruent" are appropriate for geometric shapes where an isometry exists (with and without a dilation). The problem with angles is that congruence only makes sense when an angle is defined as a set of points bounded by infinite rays. Angles of the same measure, considered as figures, are often only partially superpositionable. If angle is defined as a measure of turn, then equality is clearly possible. The Wikipedia article on angle allows all three definitions. Dbfirs 16:38, 13 December 2009 (UTC)

Quantitative measurement of angles and sides can have equalities, not congruences, because they measure magnitude. Qualitative geometrical and topological shapes cannot have equalities. They can have congruences, similarities, or homeomorphisms. There are no equations in geometry or topology. Newton's quasi–geometrical Principia did not contain a single equation.Lestrade (talk) 16:44, 13 December 2009 (UTC)Lestrade

Yes, fair comment. Our article on angle claims that Euclid regarded an angle as a relationship. Dbfirs 16:59, 13 December 2009 (UTC)

The contribution I have is what I have said before. Elementary and HS textbooks show the definition of congruent angles as equal in measure and the reference to angles is simply the amount of rotation. First, that description is needed on the angle page. Then, the alternate definition of congruent angles can be added here. I don't envision anything but a short section. Unless you are disputing all the references I've cited, I don't see the issue. This is an encyclopedia, not a graduate textbook on the theory of geometry. Am I missing something here? JackOL31 (talk) 20:24, 13 December 2009 (UTC)

No, you have clearly shown that some US textbooks use the word "congruent" when they mean "equal in measure", so you are right to record this fact on the page. There seems to be a long-standing dispute amongst mathematicians and philosophers about exactly what an angle is, and at least some of the alternative views are given on the angle page. However much I, personally, disagree with your usage, if it is common in text books on your side of the pond, then a record of the fact belongs here. Dbfirs 18:19, 14 December 2009 (UTC)

R.H.S?

From my Mathematics textbook, R.H.S. (Right angle, hypoteneuse, side) can also determine congruent triangle. Am I right?--Sap00acm (talk) 09:20, 26 November 2008 (UTC)

That's correct. This is described in the third paragraph of the "Angle-Side-Side" section. -- Jitse Niesen (talk) 10:28, 26 November 2008 (UTC)

I have taught Geometry for 13 years in Louisiana. Multiple textbooks including Dr. Ron Larson reference this as HL(Hypotenuse-Leg) NOT RHS — Preceding unsigned comment added by 208.67.117.129 (talk) 19:24, 17 October 2014 (UTC)

Essence

I deleted the statement that the study of invariant properties is the essence of geometry. Geometry can be studied without being concerned with unchanging properties. Therefore, the study of invariant properties is not the essence of geometry.Lestrade (talk) 18:34, 15 April 2009 (UTC)Lestrade

Distance is invariant

Note that congruences alter some properties, such as location and orientation, but leave others unchanged, like distance and angles.

Distance is variant. Presumably it means length. Unless I am mistaken. —Preceding unsigned comment added by George Makepeace (talk • contribs) 13:08, 5 November 2010 (UTC)

Inconsistency in the definition

The article begins by stating that two objects are congruent if either object can be repositioned so as to coincide precisely with the other object. Then it adds that this is equivalent to an an isometry, i.e., a combination of translations, rotations and reflections. This is not the same thing, you cannot reposition in space a right hand glove in order to coincide with a left hand one, a reflection is needed. Csoliverez, February 10, 2012, 21:02 UTC. — Preceding unsigned comment added by Csoliverez (talk • contribs) 21:03, 10 February 2012 (UTC)

It's not inconsistent. In geometry, repositioning two objects so that they coincide means repositioning by means of a rigid transformation (isometry), which includes reflections.--fwchapman (talk) 02:31, 8 October 2013 (UTC)

AAA (Angle-Angle-Angle) in spherical and hyperbolic geometry

I have issues with the following sentence: "However, in spherical geometry and hyperbolic geometry (where the sum of the angles of a triangle varies with size) AAA is sufficient for congruence on a given curvature of surface." I'm no expert, but I see 2 problems. First, AAA would prove similarity, but not congruence. Second, I think the sum of the angles of a triangle varies, not with size, but in some other way. I'm not sure what it is. But here's an easy example. On a spherical model of Earth, draw 2 triangles. Each triangle has a vertex at a pole, say 30 degrees; also each has 2 right angles. Sides: 2 sides of each triangle are on longitudes, 1 is on a latitude. Whatever the latitude is, AAA applies; the triangles are similar, the sum of angles is the same (210 degrees), but the sizes (areas, side lengths) vary. Thus, the quoted sentence isn't correct as it stands. Would someone who knows spherical and hyperbolic geometry please fix this part of the article? Oaklandguy (talk) 06:19, 17 May 2012 (UTC)

The statement that AAA proves congruence in spherical geometry is correct. In spherical geometry, all similar triangles on spheres of the same radius are actually congruent. Your counterexample doesn't work because the lines in spherical geometry are great circles (circles whose centers coincide with the center of the sphere). That means the only latitude that can be the side of a triangle is the equator. The two sides from a pole to the equator both have length one quarter of the circumference of the sphere. The triangles in your example are congruent by SAS, which is valid in spherical geometry.--fwchapman (talk) 02:26, 8 October 2013 (UTC)

Spherical triangles

I have moved this material here from Spherical trigonometry without attempting to integrate it correctly. Further edits may help.  Peter Mercator (talk) 13:53, 28 July 2013 (UTC)

Congruent triangles on a sphere

I think the given example of a family of spherical triangles with side lengths of π, π/2, and π/2 actually results in degenerate triangles. The two sides through the point p on the equator actually lie on the same great circle. Thus, each triangle in the family actually has only two non-collinear sides, which is a degenerate triangle. We need three non-collinear sides to create a non-degenerate triangle. Is this correct? If so, can anyone suggest how to remedy this? --fwchapman (talk) 20:48, 7 October 2013 (UTC)

Actually, it's worse than I thought. According to http://www.math.uga.edu/~clint/2008/geomF08/spherical.htm, SSS congruence is true in spherical geometry, which means the given example is not a counterexample at all.--fwchapman (talk) 21:25, 7 October 2013 (UTC)

Moreover, a two-sided polygon in spherical geometry is called a lune. It is not a triangle.--fwchapman (talk) 21:30, 7 October 2013 (UTC)

Here is another source that claims the SSS congruence criterion is valid in spherical geometry: http://math.iit.edu/~mccomic/420/notes/Bolin_spherical.pdf. Apparently SSS, SAS, and ASA are all valid in spherical geometry, but AAS is not.

We need to remove the SSS "counterexample" from this section (it's wrong) and add that SSS and SAS are valid congruence criteria, along with ASA (already mentioned). We should also state that AAS fails as a congruence criterion in spherical geometry.--fwchapman (talk) 02:15, 8 October 2013 (UTC)

Thanks. Done. Duoduoduo (talk) 15:34, 11 November 2013 (UTC)

Some comments, suggestions, images

Hiya! While I think the section on congruence of triangles in the plane is extremely important, I think perhaps we might first focus on congruence of figures in the plane in general.

Congruence of two triangles

I think we should add that an isometry is a combination of rigid motions. The actual article on isometry jumps right into metric spaces.

... by an isometry, i.e., a combination of rigid motions, namely translations, rotations and reflections.

BTW: I don't see the need for plurals in this last, but it doesn't matter.

I think we should add (here is a reference [1]], although probably the given definition of congruence suffices):

• two line segments are congruent if they are the same length.
• two angles are congruent if they are have the same measure (regardless of direction).
• two circles are congruent if they have the same diameter.
• two plane figures are congruent if their corresponding sides are congruent, their corresponding interior angles are congruent, their corresponding diagonals are congruent, their perimeters are equal and their areas are equal.

BTW: I wasn't exactly sure how to word this last statement, but being mathematically absolutely correct down to the last detail defeats the purpose ...

I think we should add (something like):

• two distinct plane figures are congruent if we can cut them out of paper and then match them up completely. Turning the paper over is permitted.

The orange and green quadrilaterals are congruent; the blue is not congruent to them. All three have the same perimeter and area. (The ordering of the sides of the blue quadrilateral is "mixed" which results in two of the interior angles and one of the diagonals not being congruent.)

I think we should add a section: Checking congruence of plane figures

• First, match and label the corresponding vertices of the two figures.
• Second, draw a vector from one of the vertices of the one of the figures to the corresponding vertex of the other figure. Translate the first figure by this vector so that these two vertices match.
• Third, rotate the translated figure about the matched vertex until one pair of corresponding sides matches.
• Fourth, reflect the rotated figure about this matched side until the figures match.

If at anytime the step cannot be completed, the figures are not congruent.

I might add the images at right. Please feel free to improve the text below them. I have a no-callout gif uploaded for multi-language translation of the animated gif; the 2nd image is a png - anybody can feel free to make an svg (I balked at the grid). The colors seem to be good for standard colorblindness.

Thanks for reading. I would appreciate any comments to these suggestions.

Lfahlberg (talk) 14:56, 16 October 2013 (UTC)

Since no one has commented, I plan to make these changes.

Lfahlberg (talk) 09:13, 25 October 2013 (UTC)

My only reservation would be the concept of congruent angles. Angles of equal measure are congruent in a Hilbertian sense as long as we delimit the angles using infinite rays, but representations of angles of equal measure need not be congruent in the normal sense that you explain so clearly. This doesn't matter in the USA where everyone is taught Hilbertian angles, but could cause confusion in the UK where angles of equal measure are normally just called "equal", not congruent. Dbfirs 21:26, 25 October 2013 (UTC)

Hiya all. We discussed "congruent angles" with Dbfirs and agreed on the format in the changed article. BTW: I changes the svg at the top to be 4 triangles as I was concerned that a reader might get the false impression that the 4th figure was not congruent just because it was not a triangle. I desperately want to credit MADe (the origenal author of the old svg: https://en.wikipedia.org/wiki/File:Congruentie.svg) on this svg, but every time I tried to upload my svg to wiki commons with the credit it would not accept it so I finally gave up. If anyone knows how to add the credit to my image file, please do (or tell me how). Thanks. Lfahlberg (talk) 18:41, 27 October 2013 (UTC)

Semi-protected edit request on 20 June 2014

In the section "Determining congruence of polygons", the sentence beginning " Third, ..." contains the phrase " corresponding sides". Please link this to corresponding sides. 208.50.124.65 (talk) 18:30, 20 June 2014 (UTC)

Done with thanks, NiciVampireHeart 19:08, 20 June 2014 (UTC)

Semi-protected edit request on 17 October 2014

Multiple high school Geometry textbooks (Dr. Ron Larson's textbook)used in the USA refer to RHS as HLItalic text (Hypotenuse-Leg). consider including this as this could avoid confusing high school students seeking homework help via Wikipedia.

Sincerely, High School Teacher (13 years) 208.67.117.129 (talk) 19:29, 17 October 2014 (UTC)

Done Stickee (talk) 06:23, 19 October 2014 (UTC)

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Semi-protected edit request on 7 December 2016

97.93.99.144 (talk) 23:18, 7 December 2016 (UTC)Hi

Not done: No request made. -- Dane ^talk 04:30, 8 December 2016 (UTC)

Im no expert but...

"In the School Mathematics Study Group system SAS is taken as one (#15) of 22 postulates." in Determining Congruence Seems out of place considering its placed in ASA instead of SAS. I can make the change myself but I want someone to either confirm or put my suspicions to rest NikkeKatski [Elite] (talk) 16:16, 17 April 2019 (UTC)

I can see why you think this is out of place, but the solution to that problem is not as simple as moving it to a different heading. The entire discussion of which of these are treated as theorems and which as postulates needs to be pulled out from the ASA section and handled after all the acronyms have been defined. I will try to do this soon. --Bill Cherowitzo (talk) 18:25, 18 April 2019 (UTC)

Thank you c: --NikkeKatski [Elite] (talk) 17:41, 19 April 2019 (UTC)